March 3rd, 2014, 11:41 AM  #1 
Newbie Joined: Mar 2014 Posts: 6 Thanks: 0  limit involving 2 sequences
Hello! I have encountered the following problem involving some limits: Let The problems asks this limit: Well I have tried several ways like Cesaro Stolz but it didn't work. After that I've tried to use a matrix:Let A be Then But I couldn't solve the problem... Any help would be appreciated! 
March 3rd, 2014, 04:33 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,821 Thanks: 1047 Math Focus: Elementary mathematics and beyond  Re: limit involving 2 sequences If you can do an series expansion at n = ? of you've got most of a proof. 
March 3rd, 2014, 09:08 PM  #3  
Newbie Joined: Mar 2014 Posts: 6 Thanks: 0  Re: limit involving 2 sequences Quote:
 
March 5th, 2014, 09:41 AM  #4 
Newbie Joined: Mar 2014 Posts: 6 Thanks: 0  Re: limit involving 2 sequences
How can I actually proof that using what you told me? Do you mean that the limit is actually e?

March 5th, 2014, 11:11 AM  #5 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,821 Thanks: 1047 Math Focus: Elementary mathematics and beyond  Re: limit involving 2 sequences
Writing out the first few terms of the series, it becomes apparent that a_n/b_n = 1, 1, 1/2, 1/3, 1/4, 1/5, 1/6 . . . We may thus write the limit as Using Stirling's approximation for n!, we have 
March 5th, 2014, 07:33 PM  #6 
Senior Member Joined: Sep 2007 From: USA Posts: 349 Thanks: 67 Math Focus: Calculus  Re: limit involving 2 sequences
Unfortunately, only testing some terms of doesn't establish proof that is is true for all values of n. Is it possible to show this limit analytically? My first few attempts tonight were unsuccessful.

March 6th, 2014, 03:12 AM  #7 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,821 Thanks: 1047 Math Focus: Elementary mathematics and beyond  Re: limit involving 2 sequences
I'm sorry, I should have mentioned that is not a complete proof. My apologies if I haven't been of much assistance.

March 6th, 2014, 06:12 AM  #8 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,821 Thanks: 1047 Math Focus: Elementary mathematics and beyond  Re: limit involving 2 sequences
I claim that Proof: Now it needs to be shown (using the induction hypothesis) that Now we may write the limit as Using Stirling's approximation for n!, we have That's about as analytic as I can get at this time and I can see why there might be problems. My logic is that, for n sufficiently large, the limit may be written as immediately above. Then, So (with L'Hopital's rule), 
March 6th, 2014, 08:10 AM  #9 
Newbie Joined: Mar 2014 Posts: 6 Thanks: 0  Re: limit involving 2 sequences
Thank you! After I wrote that I calculated that limit without using Stirling's approximation. I used d'Alembert's principle and after some arrangements I got :So this is e. Thanks again!


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